using katsurada's determination of the eisenstein series ... · title of talk: using...

Using Katsurada’s determination of the Eisenstein seriesto compute Siegel eigenforms in degree three

Cris Poor David S. YuenFordham University Lake Forest College

including slides from a book in progress withJerry Shurman, Reed College

Computational Aspects of L-functionsICERM, November 2015

Cris and David Katsurada and Eisenstein series ICERM 1 / 36

Computational Aspects of L-functions

1. Part I . What are Siegel modular forms?

2. Part II . What are examples of Siegel modular forms?

3. Part III . What good are Siegel modular forms?

4. Part IV . How are we going to compute Euler factors in degree three?

5. Part V . What Euler factors of Siegel modular forms have been seen?

6. You can see some data at: math.lfc.edu/∼yuen/genus3


Siegel Modular Forms

R ⊆ R is a commutative subring and J =[

0 I−I 0

].

General Symplectic group

GSp+n (R) = σ ∈ GL2n(R) : ∃ν ∈ R+ : σ′Jσ = νJ

Similitude: ν : GSp+n (R)→ R+ given by σ 7→ n

√det(σ)

Symplectic group

Spn(R) = ker(ν) = σ ∈ SL2n(R) : σ′Jσ = J

Siegel Upper Half Space

Hn = Ω ∈ Msymn×n(C) : Im Ω > 0



Action of σ =[A BC D

]∈ GSp+

n (R) on Ω ∈ Hn

σ · Ω = (AΩ + B)(CΩ + D)−1

Factor of Automorphy

j : GSp+n (R)×Hn → C× given by j(

[A BC D

],Ω) = det(CΩ + D)

Cocycle condition: j(σ1σ2,Ω) = j(σ1, σ2 · Ω) j(σ2,Ω)

Slash action of group on functions f : Hn → C

(f |kσ) (Ω) = ν(σ)nk−n(n+1)/2j(σ,Ω)−k f (σ · Ω)



Siegel modular group Γn = Spn(Z)

Vector space of Siegel modular forms of weight k and level one.

Mk(Γn) = holomorphic f : Hn → C : ∀γ ∈ Γn, f |kγ = f ,

and ∀Yo > 0, f is bounded on Ω : Im Ω > Yo

Siegel Phi map Φ : Mk(Γn)→ Mk(Γn−1) given by

(Φf )(Ω) = limλ→+∞

f[

Ω 00 iλ

]Siegel modular cusp forms

Sk(Γn) = ker(Φ) = f ∈ Mk(Γn) : Φf = 0


Fourier expansion

Every Siegel modular form f ∈ Mk (Γn) has a Fourier expansion

f (Ω) =∑

T :T≥0, 2T even

a(T ; f )e (〈Ω,T 〉)

• Here, e(z) = e2πiz and 〈Ω,T 〉 = tr(ΩT ).

• a (T ; Φf ) = a([

T 00 0

]; f)

Every Siegel modular cusp form f ∈ Sk (Γn) has a Fourier expansion

f (Ω) =∑

T :T>0, 2T even

a(T ; f )e (〈Ω,T 〉)


Ways to make Siegel Modular Forms

Eisenstein series

Theta series

Polynomials in the thetanullwerte

Various lifts

Specializations of symplectic embeddings.

Generating functions, multiplication, differential operators,...


Siegel Eisenstein Series

E(n)k =

∑Pn,0(Z)σ∈Pn,0(Z)\Γn

1|kσ ∈ Mk(Γn) for k > n + 1.

Pn,0(Z) = [A B0 D

]∈ Γn

Φ(E

(n)k

)= E

(n−1)k ; Φ

(E

(1)k

)= 1

k > n + 1 ensures absolute convergence on compact sets

Remarkably, the Fourier coefficients of an Eisenstein series,

a(T ;E

(n)k

), depend only upon the genus of the index T .

The algorithmic computation of the Fourier coefficients of Siegel

Eisenstein series a(T ;E

(n)k

)began with C. Siegel in the 1930s and was

completed by Hidenori Katsurada in 1999.


Example in degree n = 2

Weight 4 Eisenstein series of degree 2: z ∈ H2

E(2)4 (z) = 1 + 240 e(z22) + 2160 e(2z22) + 6720 e(3z22) + 17520 e(4z22) + · · ·

+ 13440 e〈(

112

12 1

), z〉+ 30240 e〈( 1 0

0 1 ) , z〉+ 138240 e〈(

112

12 2

), z〉

+ 181440 e〈( 1 00 2 ) , z〉+ 604800 e〈( 2 1

1 2 ) , z〉+ 362880 e〈(

112

12 3

), z〉

+ 967680 e〈(

212

12 2

), z〉+ 497280 e〈( 1 0

0 3 ) , z〉+ 1239840 e〈( 2 00 2 ) , z〉

+ 1814400 e〈( 2 11 3 ) , z〉+ · · ·

omitting GL2(Z)-equivalent terms


Type II latticesUnimodular self-dual even lattices

Definition

A lattice Λ in a euclidean space V is Type II means

Λ is even: For all u ∈ Λ, 〈u, u〉 ∈ 2Z.

Λ is self-dual: Λ = Λ∗ = u ∈ V : ∀v ∈ Λ, 〈u, v〉 ∈ Z.

For a fixed rank, necessarily a multiple of 8, the even unimodular latticesform a single genus.

rank = 8, genus = E8.rank = 16, genus = E8 ⊕ E8,D

+16. (Witt)

rank = 24, genus = 24 Niemeier lattices. (Niemeier)

rank = 32, |genus| > 80 million.


An infinite family of Type II lattices

The checkerboard lattice: Dn = v ∈ Zn :n∑

j=1

vi ≡ 0 mod 2.

Dn is even but [D∗n : Dn] = 4.

The glue vector: [1] = (1

2,

1

2, . . . ,

1

2) ∈ Qn.

The Type II lattice: D+n = Dn ∪ ([1] + Dn) .

(In fact, D+8 = E8.)


Theta series of Type II lattices

Theorem

Let Λ be a Type II lattice of rank 2k. The degree n theta series of Λ

ϑ(n)Λ (Ω) =

∑L∈Λn

e

(1

2〈LL′,Ω〉

)∈ Mk (Γn)

is a Siegel modular form of weight k and degree n.

Φ(ϑ

(n)Λ

)= ϑ

(n−1)Λ

ϑ(n)E8

= E(n)4 ∈ M4(Γn) = Cϑ(n)

E8, (Duke and Imamoglu)

ϑ(n)E8⊕E8

= ϑ(n)

D+16

if and only if n ≤ 3, (Problem of Witt)

J(4)8 = ϑ

(4)E8⊕E8

− ϑ(4)

D+16

∈ S8(Γ4) is the 1888 Schottky form. (Igusa)

The Wall:∑32

n=1 dimS16(Γn) > 80, 000, 000.


Fourier coefficients of Theta seriesGenerating functions for lattice counts are Siegel modular forms

a (T ;ϑΛ)

|AutZ(T )|= Number of sublattices Λ ⊆ Λ with Gram matrix 2T .

Example: a(

12 [ 2 1

1 2 ];ϑE8

)= 13 440 = 12 · 1120 = |AutZ [ 2 1

1 2 ]|1120

There are 1120 sublattices Λ ⊆ E8 with a basis (v1, v2) that satisfies(〈v1, v1〉〈v1, v1〉〈v2, v1〉〈v2, v2〉

)=

(2 11 2

).

There are 1120 sublattices of type A2 inside E8.

This motivates the whole theory of Siegel modular forms.


Siegel’s Theorem

Theorem (Siegel)

Let k be divisible by 4 and satisfy k > n + 1. We have∑[Λ]

1

|AutZ Λ|

E(n)k =

∑[Λ]

1

|AutZ Λ|ϑ

(n)Λ ,

where the sum is over isomorphism classes of Type II latices.

1 We can get a similar theorem for k ≡ 2 mod 4 by attachingpluriharmonic polynomials Q to the theta series

ϑ(n)Λ,Q(Ω) =

∑L∈Λn Q(L)e

(12〈LL

′,Ω〉)

but let’s skip the details.

2 The Eisenstein series is naturally associated to the Type II genus.


What good are Siegel Modular Forms?Uses in Algebraic Geometry

Satake Compactification: S (Γn\Hn) = proj (⊕∞k=0Mk(Γn))

Smooth Compactification: S (Γn\Hn) = proj (valuation subring)

The Schottky form J(4)8 = ϑ

(4)E8⊕E8

− ϑ(4)

D+16

∈ S8(Γ4) has the Jacobian locus

as its zero divisor in degree n = 4.

C4/(ΩZ4 + Z4) is the limit of Jacobians of compact Riemann surfaces of

genus 4 if and only if J(4)8 (Ω) = 0


What good are Siegel Modular Forms?They make L-functions

Both Mk(Γn) and Sk(Γn) have a basis of Hecke eigenforms.

But how are we going to compute these spaces in order to make ourL-functions?

The difficulty is in getting enough Fourier coefficients to break thespace into eigenspaces and to compute Euler factors.


The Witt MapA particular symplectic embedding.

The embedding

Wij : Hi ×Hj → Hi+j

(Ω1,Ω2) 7→[

Ω1 00 Ω2

]pulls back the the Witt Map

W ∗ij : Mk(Γi+j)→ Mk(Γi )⊗Mk(Γj)

(Ω 7→ f (Ω)) 7→ ((Ω1,Ω2) 7→ f[

Ω1 00 Ω2

])

The Witt map takes cusp forms to cusp forms.


Properties of the Witt Map

Fourier coefficients of W ∗ij f ∈ Mk(Γi )⊗Mk(Γj) in terms of f ∈ Mk(Γi+j):

a(T1 × T2;W ∗

ij f)

=∑

R∈ 12Mi×j (Z)

a([

T1 RR′ T2

]; f)

1. Pay attention to the fact that R has ij entries. Looping over theseentires is the cost of evaluating the Witt map.

2. W ∗ij ϑ

(i+j)Λ = ϑ

(i)Λ ⊗ ϑ

(j)Λ

The theta series have a beautiful decomposition under the Witt map. Thedecomposition of the Eisenstein series under the Witt map is also beautifulbut more subtle.


Garrett’s formula (c.1980)

Garrett’s Formula

For even k > 2n + 1

W ∗nnE

(2n)k =

d∑`=1

c` h`⊗h`h1, · · · , hd Hecke eigenform basis of Mk(Γn)

all c` nonzero (special values of L-functions)

(That is, E(2n)k (

(z1 00 z2

)) =

∑c`h`(z1)h`(z2) for all z1, z2)

Degree n: we don’t know the dimn d , basis h`, or special values c`. . .

. . . but all are built into only-some-of the one Siegel modular form E(2n)k of

degree 2n

And E(2n)k is computationally tractable: FCs are genus class functions

Astonishing


Plan of action

Title of talk: Using Katsurada’s determination of the Eisenstein series tocompute Siegel eigenforms in degree three.

Use the Witt pieces of Eisenstein series to span Mk(Γn)

By Garrett’s formula this will always work when k > 2n + 1.

We receive special help in degree three because Tsuyumine hascomputed dimMk(Γ3).

We need to sum over the nine entries of R ina(T1 × T2;W ∗

ij f)

=∑

R∈ 12Mi×j (Z) a

((T1 RR′ T2

); f)

.

Note we first find the sum on the right as a symbolic sum over genussymbols (in lieu of the more expensive matrix reduction).

We must have a fast way to compute FCs of Eisenstein series andKatsurada has provided it.


Eisenstein series Fourier coefficients

Siegel Eisenstein Series Fourier Coefficient Formula

For even degree n ∈ Z>0, even weight k > n + 1, definite T of rank n,write (−1)n/2 det(2T ) = DT f

2T for a fundamental discriminant DT and

fT ∈ N.

a(T ;E

(n)k

)=

2n2 L(1− (k − n

2 );χDT

)ζ(1− k)

∏ n2j=1 ζ(1− (2k − 2j))

∏p|fT

Fp(T ; pk−n−1

).

For odd degree n,

a(T ;E

(n)k

)=

2n+1

2

ζ(1− k)∏ n−1

2j=1 ζ(1− (2k − 2j))

∏2p| det(2T )

Fp(T ; pk−n−1

).

What are these Fp polynomials?


Katsurada’s 1999 article

1 An explicit formula for Siegel Series (1999) H. KatsuradaKatsurada writes recursion formulae for the Fp polynomials.

2 A mass formula for unimodular lattices with no roots (2003) O. KingKing writes a LISP program to implement Katsurada’s recursionformula for the Fp polynomials, and kindly shares it with us.

3 We modify King’s program to accept higher degrees.(So if it is wrong— we might have done it.)


Definition of Fp polynomials

For a given prime p and half-integral matrix B of degree n over Zp definethe local Siegel series by

bp(B, s) =∑

R∈Vn(Qp)/Vn(Zp)

ep (〈B,R〉) [RZnp + Zn

p : Znp]−s

ξp(B) = χp

((−1)n/2 det(2B)

).

γp(B,X ) =

(1− pn/2ξp(B)X )−1(1− X )

∏n/2i=1(1− p2iX 2), n even,

(1− X )∏(n−1)/2

i=1 (1− p2iX 2), if n is odd.

Lemma (Kitaoka)

For a nondegenerate half-integral matrix B of degree n over Zp there existsa unique polynomial Fp(B,X ) in X over Z with constant term 1 such that

bp(B, s) = γp(B, p−s)Fp(B, p−s).


Standard L-function of a Siegel Hecke eigenform

A Hecke eigenform f has L-functions of many sorts

For each p, a Satake parameter

αp = (α0,p, α1,p, · · · , αn,p) each αi ,p ∈ C

describes the eigenform behavior of f under T (p) and the Ti (p2)

Euler factor

Q(α,X ) = (1− X )n∏

i=1

(1− αiX )(1− α−1i X )

Standard L-function

Lst(f , s) =∏p

Q(αp, p−s)−1

=∏p

((1− p−s)−1

n∏i=1

(1− αi ,pp−s)−1(1− α−1

i ,p p−s)−1

)Cris and David Katsurada and Eisenstein series ICERM 24 / 36

What Euler factors did people find?Degree Two

Kurokawa discovers lifts. (1978)

L(F , s, spin) = ζ(s − k + 1)ζ(s − k + 2)L(f , s)Here eigenform f ∈ S2k−2(Γ1) and F ∈ Sk(Γ2).

Kurokawa and Skoruppa have to compute up to S20(Γ2) to find anonlift!

Congruences between lifts and nonlifts also found.


What Euler factors did people find?Degree Three

Miyawaki discovers lifts in degree three. (1992)


Miyawaki lifts

Standard L-function Euler factors sometimes decompose recognizably,showing that a Siegel modular form arises from smaller ones. Especially indegree n = 3:

Miyawaki Conjectures

1 For k even, for cusp eigenforms f ∈ S2k−4(Γ1) and g ∈ Sk(Γ1), thereexists a cusp eigenform h ∈ Sk(Γ3) such that

Lst(h, s) = L(f , s + k − 2)L(f , s + k − 3)Lst(g , s)

2 For k even, for cusp eigenforms f ∈ S2k−2(Γ1) and g ∈ Sk−2(Γ1),there exists a cusp eigenform h ∈ Sk(Γ3) such that

Lst(h, s) = L(f , s + k − 1)L(f , s + k − 2)Lst(g , s)

(1) proved by Ikeda, (2) still open



Miyawaki discovers lifts in degree three. (1992)

S12(Γ3) is one dimensional and an Ikeda-Miyawaki lift.

S14(Γ3) is one dimensional and a Miyawaki lift of type 2.

We use Katsurada’s determination of the Fourier coefficients ofEisenstein series to compute the Witt images of degree six Eisensteinseries (2010-present)

Congruences to IkedaMiyawaki lifts and triple L-values of ellipticmodular forms (2014) (Ibukiyama, Katsurada, PY).



dim S16(Γ3) = 3. Two conjugate Ikeda-Miyawaki lifts, f1 and f2 andan f3 with unimodular Satake parameters.f1 ≡ f3 modulo a prime above 107.

T f1 f3

T (2) 4414176 + 23328√

18209 −115200

T0(4) 55296(−17632637 + 1160109√

18209) −784548495360

T1(4) −4718592(−1757519 + 1503√

18209) −1062815662080

T2(4) 1207959552(−209 + 9√

18209) −352724189184

T3(4) 68719476736 68719476736



dim S18(Γ3) = 4.

Two conjugate Ikeda-Miyawaki lifts, and and two conjugate lifts of(apparently) Miyawaki’s second type.

No congruences over big primes.



dim S20(Γ3) = 6.

Three conjugate Ikeda-Miyawaki lifts, f1, f2, f3, and and two conjugatelifts f4, f5, of (apparently) Miyawaki’s second type, and one eigenformf6 with unimodular Satake parameters.

f1 ≡ f6 modulo a prime over 157

for k = 20, a standard 2-Euler factorof f6 is

1

68719476736(68719476736 + 183681155072x + 257889079808x2+

277369629719x3 + 257889079808x4 + 18681155072x5 + 68719476736x6)


What Euler factors did people find?Degree Three: this one needs verification.

dim S22(Γ3) = 9.

Three conjugate Ikeda-Miyawaki lifts, f1, f2, f3, and and threeconjugate lifts f4, f5, f6, of (apparently) Miyawaki’s second type, andthree conjugate eigenforms f7, f8, f9 with unimodular Satakeparameters.

f1 ≡ f7 modulo primes over 67 and 613

f4 ≡ f7 modulo a prime over 1753


Summary of degree n = 3At least up to weight k ≤ 22.

It looks like Miyawaki found all the possibilities for lifts in degreethree.

Can anyone prove Miyawaki’s second type of lift?

The congruence primes come from algebraic triple L-values.Thus, although not all eigenforms are lifts, so far are eigenforms areexplained by lifts.

The Kitaoka space is the subspace of Siegel modular forms whoseFourier coefficients depend only on the genus of the index. Kitaokaproved this subspace is stable under the Hecke algebra.

So far, the Eisenstein series is the only element of Mk(Γ3), fork ≤ 22, that is in the Kitaoka space.

This talk was really about computing Witt images from the Kitaoka space.


Euler factors in degree n = 4in weight k ≤ 16.

The dimensions and eigenforms for 10 ≤ k ≤ 16 were proven by P.Y. andthe 2-Euler-factors were worked out by Ryan-P.-Y.

dim S8(Γ4) = 1, (Salvati-Manni) Ikeda lift.

dim S10(Γ4) = 1. Ikeda lift.

dim S12(Γ4) = 2. One Ikeda lift, one Ikeda-Miyawaki lift.)

dim S14(Γ4) = 3. Two Ikeda lifts, one Ikeda-Miyawaki lift.

dim S16(Γ4) = 7. Two Ikeda lifts, two Ikeda-Miyawaki lifts and threeforms having the following standard 2-Euler factors, on the followingslide.


Standard Euler factors in degree n = 4in weight k = 16.

2−28(1− x)(32768− 5280x − 20755x2 − 2640x3 + 8192x4)

(8192− 2640x − 20755x2 − 5280x3 + 32768x4)

2−34(1− x)(−2048 + (−1035 + 27√

18209)x − 4096x2)

(−4096 + (−1035 + 27√

18209)x − 2048x2)

(2048 + 36x + 1601x2 + 36x3 + 2048x4)

and its conjugate

Will we see a nonlift in weight 18 if we embark on computing theeigenforms using the methods in this talk?


Thank you!


using katsurada's determination of the eisenstein series ... · title of talk: using...

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